On coefficients of Carlitz cyclotomic polynomials

Abstract

Let n ∈Z+, and Φn(x)be the nth classical cyclotomic polynomial. In [4, Theorem1], D. Lehmer showed that the geometric mean of {Φs(1) :s, n ∈Z+, s ≤n} →e ≈2.71828, as n →∞. Replacing Zby Fq[T], and the nth elementary cylcotomic polynomial Φn(x)by the Carlitz m-cyclotomic polynomial Φm(x), where m ∈Fq[T], we obtain an analogue to Lehmer’s result. We also express Φm(0) ∈F2[T]in terms of φ∗(•), the Pillai polynomial function. The resulting expression is a function field analogue of Hölder’s formula for Φn(1).